Harmonic oscillator
# Classical harmonic oscillator
Every System with a force that is linear in the excrution is a harmonic oscillator. Any conservative force can in first order be approximated as such. From e.g. Newton the equation of motion of a spring becomes $$m\frac{\text{d}^2x}{dt^2} + kx = 0$$ The general solution is $$x(t) = c_1 e^{i\omega t} + c_2 e^{-i\omega t}$$ with the frequency $\omega=\sqrt{\frac{k}{m}}$.
The classical Hamiltonian for the harmonic oscillator is given by the kinetic and potential energy $$H = \frac{1}{2}\frac{p^2}{m} + \frac{1}{2}m\omega^2x^2.$$
# Quantum mecanical harmonic oscillator
In the canonical quantisation we impose the comuutation relation $[x,p] = i$ and change variables to $$a = \sqrt{\frac{m\omega}{2}}\left(x + \frac{ip}{m\omega}\right), a^\dagger = \sqrt{\frac{m\omega}{2}}\left(x - \frac{ip}{m\omega}\right),$$ which then gives the commutator $[a,a^\dagger]=1$ and the hamiltonian $$H = \omega(a^\dagger a + \frac{1}{2}).$$
Eigenstates of the hamiltonian must be eigentstates of $a^\dagger a$ which is defined as the number operator $N$. One can derive the effect of the operators $$a^\dagger a \ket{n} = N\ket{n} = n\ket{n}$$ $$a^\dagger\ket{n} = \sqrt{n+1}\ket{n+1}$$ $$a\ket{n} = \sqrt{n}\ket{n-1}$$
In the Heisenberg picture the time evolution is absorbed into operators with the equation of motion $$i\fra¢{d}{dt}a = [a,H] = \omega a,$$ which is solved by $a(t) = e^{-i\omega t}a(0)$, which is the usual solution due to the $i$.
# Special relativity
The simplest lorentz invariant equation of motion is $$\partial^\mu\partial_\mu\phi(x,t) = \square \phi = 0.$$ This is the equation of motion for a free massless field.
The general solution is $$\phi(x,t) = \int \frac{d^3p}{(2\pi)^3} [a_p(t)e^{i\vec{p}\vec{x}}+a^*_p(t)e^{-i\vec{p}\vec{x}}],$$ with the condition $(\partial^2_t + \vec{p}\vec{p})a_p(t)=0$. This condition is the equation of motion for a harmonic oscillator. The general solution can also be rewritten in terms of four vectors when factorizing the time evolution from $a$.